A Study on Rough, Fuzzy and Rough Fuzzy Bi-ideals of Ternary Semigroups

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A Study on Rough, Fuzzy and Rough Fuzzy

Bi-ideals of Ternary Semigroups

Sompob Saelee and Ronnason Chinram

Abstract—A ternary semigroup is a nonempty set together with a ternary multiplication which is associative. Any semi-group can be reduced to a ternary semisemi-group but a ternary semigroup does not necessarily reduce to a semigroup. The notion of fuzzy sets was introduced by Zadeh in 1965 and that of rough sets by Pawlak in 1982. Applications of the fuzzy set theory and rough set theory have been found in various fields. The theory of fuzzy sets and rough sets were studied in various kinds of algebraic systems. In this paper, we study rough, fuzzy and rough fuzzy bi-ideals of ternary semigroups.

Index Terms—bi ideals, rough bi-ideals, fuzzy bi-ideals, rough fuzzy bi-ideals, ternary semigroups.

I. INTRODUCTION

T

HE formal definition of a ternary algebraic structure was given by Lehmer [13] in 1932, but earlier such struc-tures were studied by Kasner [9] and Pr¨ufer [18]. Lehmer investigated certain triple systems called triplexes which turn out to be commutative ternary group. The notion of ternary semigroups was known to Banach (cf. [14]) who is credited with an example of a ternary semigroup which does not reduce to a semigroup. A nonempty setT is called aternary semigroupif there exists a ternary operationT×T×T →T,

written as (x1, x2, x3) 7→ x1x2x3 satisfying the following

identity for anyx1, x2, x3, x4, x5∈T,

(x1x2x3)x4x5=x1(x2x3x4)x5=x1x2(x3x4x5).

Any semigroup can be reduced to a ternary semigroup but a ternary semigroup does not necessarily reduce to a semigroup, for example, Z−

is a ternary semigroup while

Z−

is not a semigroup under the multiplication over integers. In [14], Los proved that every ternary semigroup can be embedded in a semigroup.

LetT be a ternary semigroup. For nonempty subsetsA, B

andC of T,let ABC :={abc|a∈A, b∈B andc∈C}.

A nonempty subsetSof T is called aternary subsemigroup

if SSS⊆S. In [20], Sioson studied ideal theory in ternary semigroups. A nonempty subsetAof a ternary semigroupT

This work was supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

Sompob Saelee, Department of Mathematics and Statistics, Faculty of Sci-ence, Prince of Songkla University, Hat Yai, Songkhla, 90110 THAILAND e-mail: pob lee@hotmail.com (Corresponding Author).

Ronnason Chinram, Department of Mathematics and Statistics, Faculty of Science, Prince of Songkla University, Hat Yai, Songkhla, 90110 THAI-LAND and Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, THAILAND, e-mail:ronnason.c@psu.ac.th.

is called a left ideal of T if T T A⊆A, a right ideal of T

if AT T ⊆A and a lateral ideal of T if T AT ⊆ A. If A

is a left, right and lateral ideal of T, A is called an ideal

of T. A ternary subsemigroup B of T is called a bi-ideal

of T if BT BT B ⊆B. A ternary subsemigroup Qof T is called aquasi-ideal of T if QT T ∩T QT∩T T Q⊆Q and

QT T∩T T QT T∩T T Q⊆Q. The intersection of a left ideal, a right ideal and a lateral ideal is a quasi-ideal and every quasi-ideal contains in this way [20]. Later, ideal theory for ternary semigroups has studied by Dixit and Dewan. They studied quasi-ideals and bi-ideals in ternary semigroups [5] and minimal quasi-ideals in ternary semigroups [4]. In [5], Dixit and Dewan proved that every quasi-ideal of T is a bi-ideal of T but the converse is not true in general by giving example. Later, ternary semigroups were studied by some author, for example, Shabir and Bano [21] studied prime bi-ideals in ternary semigroups and Iampan [7] studied filters in ternary semigroups, etc. Recently, Santiago and Bala [19] studied regular ternary semigroups and cover semigroups of ternary semigroups.

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Kuroki [12] and Xiao and Zhang [22]. Moreover, they studied fuzzy ideals of ternary semigroups analogous to that of semigroups.

In this paper, we study rough, fuzzy and rough fuzzy bi-ideals of ternary semigroups.

II. ROUGH BI-IDEALS OF TERNARY SEMIGROUPS

Kar and Maity [8] studied congruences on ternary semi-groups. Congruence is a special type of equivalence relation which plays a vital role in the quotient structure of algebraic structures. Let T be a ternary semigroup. A congruence ρ on T is an equivalence relation on T such that for all

a, b, x, y∈T,

(a, b)∈ρimplies(xya, xyb),(xay, xby),(axy, bxy)∈ρ.

Fora∈T, the ρ-congruence class containing adenoted by

[a]ρ. A congruenceρofT is calledcompleteif[a]ρ[b]ρ[c]ρ= [abc]ρ for alla, b, c∈T. Let ρbe a congruence onT andA

a nonempty subset ofT. The sets

ρ−(A) ={x∈T |[x]ρ⊆A} and ρ−

(A) ={x∈T |[x]ρ∩A6=∅}

are called the ρ-lower and ρ-upper approximations of A, respectively.

Example II.1. Define a relation ρ on a ternary semigroup

Z−

under the usual multiplication by

xρy↔3|x−y for allx, y∈Z−

.

Then ρ is a congruence on Z−

and Z−_/ρ ₌

{[−1]ρ,[−2]ρ,[−3]ρ}. Let A = {−3,−6}. We have that ρ−(A) =∅andρ

−

(A) = [−3]ρ.

This proposition is similar to the proof of Theorem 2.1 in Kuroki [12].

Proposition II.1. ([17]) Let ρ and λ be congruences on a ternary semigroup T and A and B nonempty subsets of T. The following statements are true.

(1) ρ−(A)⊆A⊆ρ−(A).

(2) ρ−

(A∪B) =ρ−

(A)∪ρ−

(B).

(3) ρ−(A∩B) =ρ−(A)∩ρ−(B).

(4) A⊆B impliesρ−(A)⊆ρ−(B).

(5) A⊆B impliesρ−

(A)⊆ρ−

(B).

(6) ρ−(A)∪ρ−(B)⊆ρ−(A∪B).

(7) ρ−₍_A_∩_B₎_⊆_ρ−₍_A₎_∩_ρ−₍_B₎_.

(8) ρ⊆λimpliesλ−(A)⊆ρ−(A).

(9) ρ⊆λimpliesρ−₍_A₎_⊆_λ−₍_A₎_.

Proposition II.2. ([17])Letρbe a complete congruence on a ternary semigroup T and A, B and C nonempty subsets of T. Then (1) ρ−₍_A₎_ρ−₍_B₎_ρ−₍_C₎ _⊆ _ρ−₍_ABC₎ _{and (2)}

ρ−(A)ρ−(B)ρ−(C)⊆ρ−(ABC).

A nonempty subsetA of a ternary semigroupT is called a ρ-upper rough bi-ideal of T if ρ−₍_A₎

is a bi-ideal of T

andAis called a ρ-lower rough bi-idealof T ifρ−(A)is a

bi-ideal of T.

Lemma II.3. ([17]) Let ρ be a congruence on a ternary semigroup T and A a nonempty subset of T. If A is a ternary subsemigroup ofT, thenAis aρ-upper rough ternary subsemigroup ofT.

Theorem II.4. Letρbe a congruence on a ternary semigroup T and A a nonempty subset of T. If A is a bi-ideal of T, thenA is aρ-upper rough bi-ideal ofT.

Proof. Assume A is a bi-ideal of T. Then ρ−₍_A₎ ₆₌

∅. By Lemma II.3, we have ρ−₍_A₎_ρ−₍_A₎_ρ−₍_A₎ _⊆

ρ−₍_A₎

. By Theorem II.2 and Proposition II.1(5), we have

ρ−₍_A₎_{T ρ}−₍_A₎_{T ρ}−₍_A_{) =}_ρ−₍_T₎_ρ−₍_A₎_ρ−₍_T₎_ρ−₍_A₎_ρ−₍_A₎

⊆ρ−₍_{AT AT A}₎_⊆_ρ−₍_A₎

.

However, the converse of this theorem is not true in general. For example, we can see in Example II.1, ρ−

(A)

is a bi-ideal of Z−

butA is not.

Theorem II.5. Letρbe a complete congruence on a ternary semigroup T and A a nonempty subset of T such that ρ−(A) 6= ∅. If A is a bi-ideal of T, then A is a ρ-lower

rough bi-ideal ofT.

Proof. The proof of this theorem is similar to the proof of

Theorem II.4

Proposition II.6. Let T be a ternary semigroup and ρ a complete congruence on T. Then T /ρ = {[a]ρ | a ∈ T}

is a ternary semigroup under the ternary operation by

[a]ρ[b]ρ[c]ρ= [abc]ρ for all a, b, c∈T.

Let ρ be a complete congruence on a ternary semigroup

T. The ternary semigroup T /ρis called a quotient ternary semigroup of T by a congruenceρ.

Let ρ be a congruence on a ternary semigroup T. The

ρ-lower andρ-upper approximations can be presented in an equivalent form as shown below:

ρ−(A)/ρ={[x]ρ ∈T /ρ|[x]ρ⊆A}and ρ−

(A)/ρ={[x]ρ∈T /ρ|[x]ρ∩A6=∅},

respectively. Now we discuss these sets as subsets of a quotient ternary semigroupT /ρ.

Lemma II.7. ([17]) Let ρ be a complete congruence on a ternary semigroup T. IfA is a ternary subsemigroup of T, thenρ−₍_A₎_/ρ_{is a subsemigroup of}_{T /ρ}_.

Theorem II.8. Letρbe a complete congruence on a ternary semigroup T. If A is a bi-ideal of T, then ρ−

(A)/ρ is a bi-ideal of T /ρ.

Proof. Assume A is a bi-ideal of T. By Lemma II.7, we have ρ−₍_A₎_/ρ

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[x]ρ,[y]ρ,[z]ρ ∈ ρ−(A)/ρ and[a]ρ,[b]ρ ∈ T /ρ. Then there

exist x′ _∈ _[_x_]

ρ ∩ A, y′ ∈ [y]ρ ∩ A, z′ ∈ [z]ρ ∩A and a′ _∈ _[_a_]

ρ, b′ ∈ [b]ρ. Since ρ is complete, x′a′y′b′z′ ∈ [x]ρ[a]ρ[y]ρ[b]ρ[x]ρ = [xaybz]ρ. Since A is a bi-ideal of T, x′_a′_y′_b′_z′ _∈ _A

. Then [xaybz]ρ ∩ A 6= ∅. Hence [x]ρ[a]ρ[y]ρ[b]ρ[z]ρ∈ρ−(A)/ρ. Lemma II.9. ([17]) Let ρ be a complete congruence on a ternary semigroup T. IfA is a ternary subsemigroup of T, thenρ−(A)/ρis a ternary subsemigroup of T /ρ.

Theorem II.10. Letρbe a complete congruence on a ternary semigroup T and A a nonempty subset of T such that ρ−(A)/ρ 6= ∅. If A is a bi-ideal of T, then ρ−(A)/ρ is

a bi-ideal ofT /ρ.

Proof. Assume A is a bi-ideal of T. By Lemma II.9, we have ρ−(A)/ρ is a ternary subsemigroup of T /ρ. Let

[x]ρ,[y]ρ,[z]ρ ∈ ρ−(A)/ρ and [a]ρ,[b]ρ ∈ T /ρ. Then [x]ρ ⊆A,[y]ρ ⊆A and[z]ρ ⊆A. Since A is a bi-ideal of T,[x]ρ[a]ρ[y]ρ[b]ρ[z]ρ⊆A. Therefore[x]ρ[a]ρ[y]ρ[b]ρ[z]ρ∈

ρ−(A)/ρ.

III. FUZZY BI-IDEALS OF TERNARY SEMIGROUPS

First, we recall the definition of fuzzy subsets. LetT be a ternary semigroup. A function f fromT to the unit interval

[0,1]is called afuzzy subsetofT. The ternary semigroupT

itself is a fuzzy subset ofT such thatT(x) = 1for allx∈T, denoted also by T. IfA⊆T, thecharacteristic functionfA

of Ais a fuzzy subset ofT defined as follows:

fA(x) =

(

1 if x∈A,

0 if x /∈A.

Now we study fuzzy bi-ideals of ternary semigroups. Let

T be a ternary semigroup. A fuzzy subsetf of T is called afuzzy bi-idealof T iff(abc)≥min{f(a), f(b), f(c)} and

f(abcde)≥min{f(a), f(c), f(e)} for alla, b, c, d, e∈T.

Theorem III.1. Let T be a ternary semigroup and A a nonempty subset ofT. ThenAis a bi-ideal ofT if and only if fA is a fuzzy bi-ideal of T.

Proof. Assume A is a bi-ideal of T. Let a, b, x, y, z ∈ T.

Case 1: x, y, z∈A. SinceAis a bi-ideal ofT,xyz, xaybz ∈ A. Therefore fA(xyz) = 1 ≥ min{fA(x), fA(y), fA(z)}

andfA(xaybz) = 1≥min{fA(x), fA(y), fA(z)}.

Case 2:x /∈A ory /∈A or z /∈A. ThusfA(x) = 0 or fA(y) = 0or fA(z) = 0. Hence min{fA(x), fA(y), fA(z)} = 0 ≤ fA(xyz) and min{fA(x), fA(y), fA(z)} = 0 ≤

fA(xaybz).

LetT be a ternary semigroup. A nonempty subsetS ofT

is called a prime subsetof T if for all x, y, z∈T, xyz∈S

implies x ∈ S or y ∈ S or z ∈ S. A bi-ideal S of T is called a prime bi-ideal of T if S is a prime subset of T. A fuzzy subset f of T is called a prime fuzzy subset of T

if f(xyz) ≤max{f(x), f(y), f(z)} for all x, y, z ∈ T. A fuzzy bi-idealf ofT is called aprime fuzzy bi-idealofT if

f is a prime fuzzy subset ofT.

Lemma III.2. ([17])LetT be a ternary semigroup andAa nonempty subset ofT. ThenAis a prime subset ofT if and only if fA is a prime fuzzy subset of T.

Theorem III.3. Let T be a ternary semigroup and A a nonempty subset of T. Then A is a prime bi-ideal of T if and only iffA is a prime fuzzy bi-ideal ofT.

Proof.It follows from Lemma III.2 and Theorem III.1.

Letf be a fuzzy subset of a set (a ternary semigroup) T. For any t∈[0,1], the set

ft={x∈T |f(x)≥t} andfts={x∈T |f(x)> t}

are called at-levelsetand at-strong levelsetoff, respectively [22].

Theorem III.4. Letf be a fuzzy subset of a ternary semi-group T. Then f is a fuzzy bi-ideal of T if and only if for allt∈[0,1],if ft6=∅, thenftis a bi-ideal of T.

Proof. Assume f is a fuzzy bi-ideal of T. Let t ∈ [0,1]

such that ft 6= ∅. Let x, y, z ∈ ft. Then f(x) ≥ t, f(y) ≥ t and f(z) ≥ t. Since f is a fuzzy bi-ideal of

T, f(xyz) ≥ min{f(x), f(y), f(z)} ≥ t and f(xaybz) ≥

min{f(x), f(y), f(z)} ≥ t for all a, b ∈ T. Therefore

xyz, xaybz ∈ ft. Hence ft is a bi-ideal of T.

Con-versely, assume for all t ∈ [0,1], if ft 6= ∅, then ft

is a bi-ideal of T. Let a, b, x, y, z ∈ T. Choose t =

min{f(x), f(y), f(z)}. Then x, y, z∈ ft. This implies that ft 6= ∅. By assumption, we have ft is a bi-ideal of T. So xyz, xaybx∈ft. Therefore f(xyz)≥t andf(xaybz)≥t.

Hence f(xyz) ≥ min{f(x), f(y), f(z)} and f(xaybz) ≥

min{f(x), f(y), f(z)}.

Lemma III.5. ([17]) Let f be a fuzzy subset of a ternary semigroup T. Then f is a prime fuzzy subset of T if and only if for all t∈[0,1], ifft6=∅, thenft is a prime subset

ofT.

Theorem III.6. Letf be a fuzzy subset of a ternary semi-groupT. Thenf is a prime fuzzy bi-ideal ofT if and only if for all t∈[0,1], ifft6=∅, thenftis a prime bi-ideal of T.

Theorem III.7. Letf be a fuzzy subset of a ternary semi-group T. Then f is a fuzzy bi-ideal of T if and only if for allt∈[0,1],if fs

t 6=∅, thenfts is a bi-ideal ofT.

Proof. The proof of this theorem is similar to the proof of

Theorem III.4

Theorem III.8. Letf be a fuzzy subset of a ternary semi-group T. Thenf is a prime bi-ideal of T if and only if for allt∈[0,1],if fs

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IV. ROUGH FUZZY BI-IDEALS OF TERNARY SEMIGROUPS

Let f be a fuzzy subset of a ternary semigroup T and ρ

be a congruence onT. Then the fuzzy setsρ−₍_f₎

andρ−(f)

are defined by

ρ−

(f)(x) = sup a∈[x]ρ

f(a)andρ−(f)(x) = inf

a∈[x]ρ

f(a)

are called theρ-upperandρ-lower approximationsof a fuzzy setf, respectively [6].

Example IV.1. Define a relation ρ on a ternary semigroup

Z−

under the usual multiplication by

xρy↔2|x−y for alla, b∈Z−

.

It is easy to see thatρis a congruence onZ−

. Letf(x) = 1

−x

for allx∈Z−

. Then

ρ−(f)(x) = 0for allx∈Z −

and

ρ−

(f)(x) =

  

1 if x∈Z−

is odd,

1

2 if x∈Z

−

is even.

Lemma IV.1. ([17]) Let ρ be a congruence on a ternary semigroup T, f a fuzzy subset of T and t ∈[0,1], then (1)

(ρ−(f))t=ρ−(ft)and (2)(ρ−(f))st=ρ−(fts).

Theorem IV.2. Letρbe a complete congruence on a ternary semigroup T. If f is a fuzzy bi-ideal ofT, then ρ−

(f) and ρ−(f)are fuzzy bi-ideals ofT.

Proof. It follows from Theorem III.4, Theorem III.7, Theo-rem II.4, TheoTheo-rem II.5 and Lemma IV.1.

Note that if ρ−₍_f₎

and ρ−(f) are fuzzy bi-ideals of a

ternary semigroup T, in general f need not be a fuzzy bi-ideal ofT. For example, we can see in Example IV.1,ρ−(f)

is a fuzzy bi-ideal ofT butf is not.

V. PROBLEMS OF HOMOMORPHISMS

LetT1 andT2 be ternary semigroups. A mappingϕfrom

T1toT2is called ahomomorphismfromT1toT2ifϕ(abc) =

ϕ(a)ϕ(b)ϕ(c)for alla, b, c∈T1. Then the setκ={(a, b)∈

T1×T1 |ϕ(a) = ϕ(b)} is called the kernel of ϕ. We have

that κis a congruence onT1.

Lemma V.1. ([17]) Let ϕ be an onto homomorphism from a ternary semigroup T1 to a ternary semigroup T2, ρ2 a

congruence onT2,ρ1={(x, y)∈T1×T1|(ϕ(x), ϕ(y))∈

ρ2}andAa nonempty subset ofT1. The following statements

are true.

(1) ρ1 is a congruence onT1.

(2) If ρ2 is complete andϕis 1-1, thenρ1 is complete.

(3) ϕ(ρ−

1(A)) =ρ

−

2(ϕ(A)).

(4) ϕ(ρ1−(A))⊆ρ2−(ϕ(A)).

(5) If ϕis 1-1, thenϕ(ρ1−(A)) =ρ2−(ϕ(A)).

Lemma V.2. ([17]) Let ϕ be an onto homomorphism from a ternary semigroup T1 to a ternary semigroup T2, ρ2 a

congruence onT2,ρ1={(x, y)∈T1×T1|(ϕ(x), ϕ(y))∈

ρ2}andAa nonempty subset ofT1. Thenρ−1(A)is a ternary

subsemigroup of T1 if and only if ρ−2(ϕ(A)) is a ternary

subsemigroup ofT2.

Theorem V.3. Let ϕ be an onto homomorphism from a ternary semigroupT1 to a ternary semigroupT2,ρ2 a

con-gruence onT2,ρ1={(x, y)∈T1×T1|(ϕ(x), ϕ(y))∈ρ2}

andAa nonempty subset ofT1. Thenρ−1(A)is a bi-ideal of

T1 if and only if ρ−2(ϕ(A))is a bi-ideal ofT2.

Proof. Assume ρ−

1(A) is a bi-ideal of T1. By Lemma

V.2, we have ρ−

2(ϕ(A)) is a ternary subsemigroup

of T2. ϕ(ρ−1(A))ϕ(T1)ϕ(ρ

−

1(A))ϕ(T1)ϕ(ρ

−

1(A)) =

ϕ(ρ−

1(A)T1ρ−1(A)T1ρ−1(A)) ⊆ ϕ(ρ

−

1(A)). By Lemma

V.1(3), ρ−

2(ϕ(A))ρ

−

2(T2)ρ

−

2(ϕ(A))ρ

−

2(T2)ρ

−

2(ϕ(A)) ⊆

ρ−

2(ϕ(A)). So ρ

−

2(ϕ(A)) is a bi-ideal of T2. The proof of

the converse is similar.

Lemma V.4. ([17])Letϕbe an isomorphism from a ternary semigroup T1 to a ternary semigroup T2,ρ2 a congruence

on T2,ρ1 ={(x, y)∈T1×T1 |(ϕ(x), ϕ(y))∈ρ2} and A

a nonempty subset of T1. Then ρ−1(A) is a prime subset of

T1 if and only if ρ−2(ϕ(A))is a prime subset ofT2.

Theorem V.5. Let ϕ be an isomorphism from a ternary semigroup T1 to a ternary semigroup T2,ρ2 a congruence

a nonempty subset ofT1. Thenρ−1(A)is a prime bi-ideal of

T1 if and only if ρ−2(ϕ(A))is a prime bi-ideal ofT2.

Proof.It follows from Lemma V.4 and Theorem V.3.

on T2, ρ1 = {(x, y) ∈ T1 ×T1 | (ϕ(x), ϕ(y)) ∈ ρ2}

and A a nonempty subset of T1. Then ρ1−(A)is a ternary

subsemigroup of T1 if and only if ρ2−(ϕ(A)) is a ternary

subsemigroup ofT2.

a nonempty subset of T1. Thenρ1−(A)is a bi-ideal ofT1 if

and only ifρ2−(ϕ(A))is a bi-ideal ofT2.

Proof.By Lemma V.1(5), we have ρ1−(A) =ρ2−(ϕ(A)).

The proof of this theorem is similar to the proof of

TheoremV.3.

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a nonempty subset of T1. Thenρ1−(A)is a prime subset of

T1 if and only if ρ2−(ϕ(A))is a prime subset ofT2.

a nonempty subset of T1. Then ρ1−(A)is a prime bi-ideal

ofT1 if and only ifρ2−(ϕ(A))is a prime bi-ideal ofT2.

Proof.It follows from LemmaV.8 and TheoremV.7.

We can obtain the following conclusion easily in a quotient ternary semigroup.

Corollary V.10. Let ϕ be an isomorphism from a ternary semigroupT1to a ternary semigroupT2,ρ2a complete

con-gruence onT2, ρ1={(x, y)∈T1×T1|(ϕ(x), ϕ(y))∈ρ2}

and A a nonempty subset of T1. The following statements

are true.

(1) ρ1−(A)/ρ1 is a bi-ideal of T1/ρ1 if and only if

ρ2−(ϕ(A))/ρ2 is a bi-ideal of T2/ρ2.

(2) ρ−

1(A)/ρ1 is a bi-ideal of T1/ρ1 if and only if

ρ−

2(ϕ(A))/ρ2 is a bi-ideal of T2/ρ2.

(3) ρ1−(A)/ρ1 is a prime bi-ideal ofT1/ρ1 if and only if

ρ2−(ϕ(A))/ρ2 is a prime bi-ideal of T2/ρ2.

(4) ρ−

1(A)/ρ1 is a prime bi-ideal of T1/ρ1 if and only if

ρ−

2(ϕ(A))/ρ2 is a prime bi-ideal of T2/ρ2.

ACKNOWLEDGMENT

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thai-land.

The authors are grateful to referees for the valuable com-ments and suggestions.

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Sompob Saelee was born in Hat Yai, Thailand, in 1987. He finished his B.Sc. from Prince of Songkla University in 2008. Now he is continuing his graduation in Master degree at same university. Both of Bachelor and Master level he got a Science Achievement Scholarship of Thailand to support his education.